Electromagnetic Scoot Samuel E. Gralla 1 and Kunal Lobo 1 1 Department of Physics, University of Arizona, Tucson, Arizona 85721, USA Recent work on scattering of massive bodies in general relativity has revealed that the mechanical center of mass of the system (or, more precisely, its relativistic mass moment ) undergoes a shift during the scattering process. We show that the same phenomenon occurs in classical scattering of charged particles in flat spacetime and study the effect in detail. Working to leading order in the interaction, we derive formulas for the initial and final values of the mechanical and electromagnetic energy, momentum, angular momentum, and mass moment. We demonstrate that the change in mechanical mass moment is balanced by an opposite change in the mass moment stored in the electromagnetic field. This is a non-radiative exchange between particles and field, analogous to exchange of kinetic and potential energy. A simple mechanical analogy is a person scooting forward on the floor, who exchanges mass moment with the floor. We therefore say that electromagnetic scattering results in an electromagnetic scoot. I. INTRODUCTION Scattering experiments, whether real or fictitious, offer a simple way to gain understanding of the physical impli- cations of a theory. The small-deflection limit provides a further simplified testing ground where precise analyti- cal results are usually possible. The study of small-angle scattering in general relativity began in the 1980’s with the derivation of the first-order [1] and second-order [2] deflection angle, and has recently seen a resurgence of in- terest due to connections with quantum scattering meth- ods and the dynamics of bound systems. The new com- putational firepower thrown at this problem has resulted in spectacular progress (with references too numerous to list here), with the latest results now probing the fourth order in the small-angle approximation [3, 4]. Inspired by this rich, interconnected set of results, we set about to understand gravitational scattering with a new approach using self-force methods [5]. We managed to reproduce the second-order results from the 1980s, but discovered, to our surprise, an overlooked feature of the problem that appears even at the first order beyond straight line motion. In addition to computing the en- ergy, momentum, and angular momentum of the parti- cles, we considered the last, overlooked conserved quan- tity: the mass moment. For a system of point particles, the mass moment is defined by N mech =  I E I r I − t  I p I , (1) where r I , E I and p I are the position, energy and mo- mentum (respectively) of the particles labeled by I . 1 We include the subscript “mech” to emphasize that any field contributions have not been included in this formula. 1 We set the speed of light to unity (c = 1) and regard relativistic mass and energy as equivalent. If we had not made this choice, we would divide the first term of (1) by c 2 , ensuring that mass moment has units of mass times length. The mass moment is the position-weighted energy of the system minus its total momentum times time, and its conservation reflects the uniform motion of the cen- ter of energy. It is numerically equal to the total energy times the center of energy at time t = 0, and it thereby tracks the position of the center of energy at a fiducial time. Although the total value can always be set to zero by a translation, the mass moment is additive (unlike the center of energy) and can therefore be budgeted like the energy, momentum, and angular momentum. That is, we can ask about exchange of mass moment between different degrees of freedom, or radiation of mass mo- ment away to infinity. From a relativistic point of view, mass moment is inseparable from the angular momen- tum, since the two mix under boosts and only together form a relativistically invariant object (e.g., [6]). In the scattering of two masses m 1 and m 2 , the impor- tant mass scales are the initial total energy E 0 and the relativistic reduced mass µ, E 0 =  m 2 1 + m 2 2 +2γm 1 m 2 , µ = m 1 m 2 E 0 , (2) where γ is the initial relative Lorentz factor. In the small- deflection limit, the center of energy-momentum (CEM) frame scattering angle is proportional to χ = GE 0 bv 2 ≪ 1, (3) where G is Newton’s constant, b is the impact param- eter, and v is the initial relative velocity. In our study of small-angle gravitational scattering through second or- der in χ [5], we found that the mechanical mass moment changes during the scattering process. At leading order in χ, the change is ΔN mech =2µbχγ (1 − 3v 2 ) log m 2 + γm 1 m 1 + γm 2 ˆ z, (4) where ˆ z is a unit vector pointing from particle 1 to par- ticle 2 at early times. This form makes clear that the effect disappears in the Newtonian limit v → 0, as it must. However, plugging in for χ shows that the change arXiv:2112.01729v1 [gr-qc] 3 Dec 2021